I wouldn't say totally failed. You killed the third and it only cost you zealots, which you're expected to lose like that anyways. When doing a pressure like that, you should really have a pylon up in advance so your (ideally) first warp in is right there.

There were broodlords/infestors that game? Did I even watch the right game? Well nonetheless, in this game you pretty much lost to mutas alone.

I mostly had a problem with your strategy:

The FEE sim city was questionable, but not a big deal. I don't really know specific build orders that well but I felt your second gas was a bit too early (may be wrong) for a non-tech build, but your third and fourth gas were definitely too early for no tech. Less gas generally means better economy with minerals being spent on workers, expansions and infrastructure where as a lot of gas suggests heavy teching like colossi, stargate and to a lesser extent upgrades. So taking four gasses before optimally saturated on two base only to go 5gate with observer is pretty unnecessary because your gas needs are met with only two geysers.

For one thing, 5gate with observer isn't enough production on two bases. You can support that before you're optimally saturated and once you get into the 40-50 worker range you have more money than you can spend and float resources. You start floating money as early as 8:00, which could have been spent on more gates, a third, tech, or upgrades. Basically, as you get more mining bases, the more income you get, which means the more production you need. As you start to float resources you should really consider adding on more warpgates until you can effectively spend your money. Despite the mediocre start, at the 14:00 mark if you could have spend all your resources you would have a reasonable enough army to try to fight/kill him.

The Zerg plays with a more advanced mindset. He isn't trying to just straight up kill you, he's expanding quickly while trying to slow your economy. After killing the third you are unable to put on any real pressure. For how long you stay on two base to his three and eventually four, you either need to deal significant damage, slow him down enough so your economy is equal or expand at a similar rate. What this boils down to is either: attack him and deal damage or take a third and defend until you can max out. It's pretty easy for mediocre Zergs to die to a 200/200 protoss army with upgrades and tech.

A note specific this game: The micro at the fight in middle was pretty bad. You need to forcefield to protect your ranged units and let your zealots kill lings and focus fire mutas with your stalkers. If your opponent commits to mutas like he did (more than 10-15 basically) then the easiest counter is just to get a few archons to melt ling/muta. If you spend your money well, had archons and micro half decently you could easily handle his army despite the economic disadvantage.

While I'm a gimmicky Protoss player, I feel you play similar to me. For PvZ I would look into a variety of 6gate style builds. Things like: hard 6gate timing, 6gate +1 blink (transitioning to a fast +3), 4-6gate into fast third and a similar 4gate with colossi (a good way to spend gas if you take four quick gas). When playing focus on timing attacks (warping in from 5-6 gates and chrono them for more units before you attack) and being able to take a quicker third if your initial pressure fails and being able to defend it.